10/10/2016 Biricha Digital Power Ltd http://www.biricha.com/articles/view/power_supply_compensator_types 1/8 Foundations (Part 2.A) - Common Power Supply Compensators and Their Transfer Functions tags: compensators, transfer functions, poles zeros Dr. Ali Shirsavar, Dr. Michael Hallworth Introduction In the previous articles we covered all the basic foundation material needed to understand the fundamentals of power supply compensator design. In this article we will discuss commonly used compensators in power supplies. We will explain their circuits, transfer functions and their poles and zeros. In the majority of the cases, analog compensators for PSUs, take the form of an inverting op-amp which is usually internal to the controller IC, plus some external capacitors and resistors. Of course the capacitors and resistors determine the position of the poles and zeros based on the transfer function of the op-amp circuit. Building on the material covered in the previous articles, we are now in an ideal position to cover this topic. One thing to remember about PSU compensators is that there is nothing scary about them. We are simply dealing with an inverting op-amp with some impedances, just like what we studied in our very rst few lectures at university. Consider the generic inverting op-amp in Figure 1. Figure 1. Generic inverting op-amp.
Foundations (Part 2.A) - CommonPower Supply Compensators andTheir Transfer Functionstags: compensators, transfer functions, poles zeros
Dr. Ali Shirsavar, Dr. Michael Hallworth
IntroductionIn the previous articles we covered all the basic foundation material needed to understand thefundamentals of power supply compensator design. In this article we will discuss commonly usedcompensators in power supplies. We will explain their circuits, transfer functions and their polesand zeros.
In the majority of the cases, analog compensators for PSUs, take the form of an inverting op-ampwhich is usually internal to the controller IC, plus some external capacitors and resistors. Ofcourse the capacitors and resistors determine the position of the poles and zeros based on thetransfer function of the op-amp circuit. Building on the material covered in the previous articles,we are now in an ideal position to cover this topic.
One thing to remember about PSU compensators is that there is nothing scary about them. Weare simply dealing with an inverting op-amp with some impedances, just like what we studied inour very �rst few lectures at university.
Consider the generic inverting op-amp in Figure 1.
We know from university lectures that the transfer function of this circuits is:
Equation 1
Where Z1 and Z2 are usually a combination of capacitors and resistors. With this simple circuit wecan construct almost all of the compensators that are used in analog power supplies.
Type I CompensatorIf I now replace Z2 with a capacitor (C = 10nF) and Z1 with a resistor, (R = 1.6kΩ) I will have createdmyself a simple Type I compensator. Furthermore, by selecting the values of the capacitor and theresistor I can control the position of any poles and zeros that this compensator has. Please notethat for simplicity of teaching we are not considering the dc reference source usually connected tothe non-inverting input and the biasing resistor usually connected between the inverting inputand ground. These two elements only a�ect the dcanalysis and do not impact the ac transferfunction of our compensator; in other words they have no impact on the position of our poles andzeros
My transfer function therefore becomes:
Equation 2
My gain at a certain frequency f will be:
Equation 3
Equation 4
And, after taking into account the inverting action of the op-amp, the phase will be:
Looking at the above equations there some very important observations to point out:
We have a pole at origin as the denominator of transfer function becomes zero only when s= 0.From the gain equation, we can see that the gain at low frequencies approaches in�nity.This in general is exactly what we want in a power supply, as a high gain at low frequenciestranslates into near zero steady state error.Again from the gain equation we can see that the gain becomes 1 at f = 1/ (2 π R C), i.e.when f = 10kHz. This means that in dB world the gain plot crosses the zero dB axis. Manytext books or application note call this “the position of the pole at origin”. What they actuallymean is “the frequency at which the gain due to the pole at origin will cross the 0 dB point”.This is the terminology that we will use from now on. Hence, from now on, every time wesay “we place the pole at origin at a certain frequency,” we mean that we select the values ofresistors and capacitors such that the gain plot due to the pole at origin crosses 0 dB at acertain frequency.From the phase equation we can see that we have a phase lag of -90 . Which of course isthe characteristics of having a single pole in our system
Type I compensators are relatively rare in high performance power supplies and we have includedit here for teaching purposes and to explain the concept of the position of the pole at origin. TypeII and Type III compensators are by far more common and the vast majority of power supplies arestabilised with only just these two circuits. Let us go through them.
Type II CompensatorThis is one of the most popular compensators as it is almost always used in current mode powersupplies. We will discuss current mode control in a later article.
The circuit, including a simple buck power stage is given in �gure 2.
Figure 2. Type II compensator circuit. Please note that the PWM generator and the op-amp areusually internal to our IC.
Referring back to our previous lecture notes, we can immediately identify that we have 1 pole atorigin (ωp0), a second pole (ωp2) and a zero (ωz1). Please note that the equations are in rad/s andas always the component values determine the position of these poles and zeros:
Equation 7
It is easy to see that by appropriate selection of the component values we can place our poles andzeros in such a way to shape our Bode plots so that we meet the stability criteria as discussed inearlier articles.
You can see from the transfer function that we have one zero. This of course means that we canforce a maximum phase boost of 90 . But sometimes 90 of phase boost is just not enough; acase in point being almost all our voltage mode power supplies. We therefore need anothercompensator that allows more phase boost; this is our Type III.
Type III CompensatorThe circuit for our Type III compensators is given in �gure 3
With an addition of just an extra resistor and capacitor (R3 and C2) we can convert our Type IIcompensator into a type III and get the extra phase boost that we need. The transfer function isgiven below:
Equation 8
We can see now that again we have our desirable pole at origin to remove our steady state o�set,but this time we have an extra pole (ωp3) and more importantly and extra zero (ωz2) compared toour Type II compensator. The presence of 2 zeros of course suggest that we can have 180 ofphase boost compared to our Type II’s 90 .
The equations determining the positions of our poles and zeros are:
How Well Does the Transfer Function Matchwith RealityThe good news is that with just these two transfer functions we can pretty much stabilise most ofthe analog power supplies in the world. We will learn how to place the poles and zeros and step-by-step real life design in future articles. For now let us experiment with the values of resistorsand capacitors to place our poles and zeros and then measure a real compensator to see if itmatches with our theory.
We will use Biricha Digital’s automated power supply design software to �rst design a Type IIIcompensator, then we will build and measure with a Bode 100 network analyser. You can seefrom �gure 4 that Biricha WDS has placed the poles and zeros of the compensator and calculatedthe component values. The nearest preferred values of the components are given in the righthand column of �gure 4.
Inserting the component values in the equations of the Type III compensator above should giveexactly the same poles and zeros as calculated by WDS and shown in the �gure. Please note thatthe equations given earlier are in rad/s, but in WDS they are in Hz. For now we are not interestedin the pole zero placement strategy; we will discuss detailed design in a later article. For now weare just making sure that our equations are correct and the simulated frequency response �ts wellwith the real life one.
Figure 4. Component values and the simulated poles and zeros of a type III compensator ascalculated by WDS.
The comparison between the simulated Bode plot and the real measurement of a realcompensator with the component values of �gure 4 is shown in �gure 5. You can see that we havealmost a perfect match and therefore our equations are correct and the component valuesaccurately place the poles and zeros of the compensator exactly where we expect them to be.
All we need to do now is to devise a strategy for placing our poles and zeros such we meet ourspeci�cations and the stability criteria. We will do this, in detail in the next few articles.
Figure 5. Type III compensator Bode plot simulated by Biricha WDS vs. real measurement
Concluding RemarksIn this article, we discussed compensators, their poles and zeros, transfer functions and clari�edwhat we mean by the phrase “placing the pole at origin at a certain frequency”. We showed, onceagain, that our mathematical presentations of our circuits match almost perfectly with realmeasurements.Furthermore, we detailed all the necessary equations relating the values ofcapacitors and resistors to the position of our compensators’ poles and zeros.
We are now ready to start designing our compensators to stabilise our power supplies. In the nextfew articles we will discuss voltage mode and current mode control strategies and go through thecompensator design procedures in a step-by-step manner.
Things to Try1 – Download a copy of Biricha WDS PSU Design software (/wds)
2 – Attend one of our Analog Power Supply Design (/aps) workshops
2 – Visit OMICRON Lab website (http://www.omicron-lab.com/bode-100) for more informationabout Bode 100
Bibliography[1] Biricha Digital’s Analog Power Supply Design Workshop Manual
